In the “proof” presented, the series 1-1+1… is “shown” to equal to 1⁄2 by a particular choice of interleaving of the values in the series. But with other methods of interleaving, the sum can be made to “equal” 0, 1 1⁄3 or indeed AFAICT any rational number between 0 and 1.
So… why is the particular interleaving that gives 1⁄2 as the answer “correct”?
Interleaving isn’t really the right way of getting consistent results for summations. Formal methods like Cesaro Summation are the better way of doing things, and give the result 1⁄2 for that series. There’s a pretty good overview on this wiki article about summing 1-2+3-4.. .
In the “proof” presented, the series 1-1+1… is “shown” to equal to 1⁄2 by a particular choice of interleaving of the values in the series. But with other methods of interleaving, the sum can be made to “equal” 0, 1 1⁄3 or indeed AFAICT any rational number between 0 and 1.
So… why is the particular interleaving that gives 1⁄2 as the answer “correct”?
Interleaving isn’t really the right way of getting consistent results for summations. Formal methods like Cesaro Summation are the better way of doing things, and give the result 1⁄2 for that series. There’s a pretty good overview on this wiki article about summing 1-2+3-4.. .